HILBERT S THEOREM ON THE HYPERBOLIC PLANE

Transcription

1 HILBET S THEOEM ON THE HYPEBOLIC PLANE MATTHEW D. BOWN Abstract. We recount a proof of Hilbert s result that a complete geometric surface of constant negative Gaussian curvature cannot be isometrically immersed in Euclidean 3, and, in particular, that the hyperbolic plane can not be isometrically embedded in 3. In 1899, David Hilbert published his landmark book [1] Grundlagen der Geometrie (Foundations of Geometry), in which he gave precise axioms for Euclidean geometry and proved their consistency relative to classical mathematics. In 1901, he followed up on his work with the answer to a related question [2]: to what extent is hyperbolic geometry, relatively newfangled at the time, just Euclidean geometry? After all, spherical geometry has a model in, naturally, the two-sphere embedded in Euclidean 3. Hilbert showed, however, that hyperbolic geometry does not have a model in three-dimensional Euclidean geometry. More precisely, he proved: Theorem (Hilbert, 1901). A complete geometric surface S with constant negative curvature cannot be isometrically immersed in 3. Corollary. The hyperbolic plane H 2 cannot be isometrically embedded in 3. In this paper, we give a proof of Hilbert s Theorem, or at least most of a proof. (We sacrifice a few details here and there.) We generally follow docarmo [3], who himself follows Hilbert to a large extent. We do, however, mix in some of the ideas and intuition of Treibergs [4], with a few minor ideas of our own. We include an appendix containing some results of curve and surface geometry that we will require. 1. The Proof Let S be as in the hypothesis of Hilbert s Theorem. After multiplying the metric on S by a constant factor, we may assume that S has constant Gauss curvature K = 1. Now, for any p S, the exponential map exp p : T p S S is a local diffeomorphism, we may use it to pull back the metric on S to a metric on T p S, v, w u = d(exp p ) u (v), d(exp p ) u (w) exp p (u). (Note that completeness of S gives that exp p is defined on all of T p S.) We let S denote T p S with this metric structure. In essence, we have defined S to be locally isometric to S under the exponential map. So, if ϕ : S 3 is an isometric immersion, then so is ϕ exp p : S 3. Since S is locally isometric a manifold of constant Gauss curvature 1, it itself has constant Gauss curvature 1. Thus, to prove Hilbert s Theorem, it suffices to show that there is no isometric immersion ϕ : S 3 of a plane with constant curvature 1 into 3. We do this in eight lemmas: Lemma 1. The area of S is infinite. Proof. Since S and and the hyperbolic plane are regular surfaces with the same constant Gaussian curvature, we have from Minding s Theorem (see appendix) that they are locally isometric under some local isometry π : H 2 S. By a version of a theorem of Hadamard (appendix), the map π is a covering map. Since S is simply-connected, π is then a homeomorphism. The Inverse Function Theorem then gives that ϕ is a diffeomorphism and, thus, a global isometry. As for the area of H 2, we use the upper-half plane model in which the metric takes the form ds 2 = (dx 2 + dy 2 )/y 2 so that Area(H 2 ) = EG F 2 dy dx = H dy dx =. y2 Date: October 2,

2 2 MATTHEW D. BOWN Henceforth, we assume that there is an isometric immersion ϕ : S 3 with S a geometric surface homeomorphic to a plane and having constant negative curvature 1. Note that since ϕ is an immersion, it is locally a diffeomorphism onto its inverse (by the Inverse Function Theorem): for every p S there is a neighborhood V S such that ϕ = ϕ V : V ϕ(v ) is a diffeomorphism. Therefore, we may define asymptotic directions (see appendix) on S at p by pulling back those at ϕ(p) ϕ(v ) 3, and similarly for any other local quantity on ϕ(s ). We will therefore essentially identify S and ϕ(s ) in what follows. For instance, when the following lemma discusses the asymptotic curves of a parameterization x : U 2 S, it is really referring to the asymptotic curves of ϕ x (with the assumption that ϕ x(u) is a diffeomorphism). Ultimately, we will show that identifying S with ϕ(s ) in this way leads to a contradiction. Lemma 2. For each p S, there is a parameterization x : U 2 S with p x(u) =: V such that the coordinate curves of x are the asymptotic curves of x are the asymptotic curves of V and form a Tchebyshef net. (See the appendix for the defintion and properties of Tchebyshef nets and asymptotic curves.) Proof. Let p S and V be a neighborhood of p. Since K < 0, it can be shown that we can parameterize a neighborhood V S of p by x such that the coordinate curves of x are the asymptotic curves of V. Then, denoting the coefficients of the second fundamental form of S by e, f, and g, we have e = 0 = g. After a fairly brief and typical curves-and-surfaces sort of calculation using the fact that K = 1, we find that the normal vector N in this parameterization is parallel to x uv. (See [3], p. 448 for details.) Then E v = 2 x uv, x u = 0, G u = 2 x uv, x v = 0, which implies that the coordinate curves form a Tchebyshef net. Lemma 3. Let V S be a coordinate neighborhood of S such that the coordinate curves are the asymptotic curves in V. Then the area A of any quadrilateral formed by the coordinate curves is smaller than 2π. Proof. From the proof of Lemma 2, a parameterization whose coordinate curves are asymptotic curves actually form a Tchebyshef net. Then, from the facts about Tchebyshef nets cited in the appendix, we can reparameterize to coordinates (u, v) in which the coefficients of the first fundamental form are given by E = 1 = G, F = cos θ. Let be a quadrilateral formed by the coordinate curves with vertices (u 1, v 1 ), (u 2, v 1 ), (u 2, v 2 ), (u 1, v 2 ) and interior angles α 1, α 2, α 3, α 4. Note that α i < 2π for each i since the asymptotic directions are linearly independent. From (2) of the appendix with K = 1, we have that θ uv = sin θ. Putting all this together, we get A = EG F 2 du dv = sin θ du dv = θ uv du dv = (θ v (u 2, v) θ v (u 1, v)) dv = θ(u 2, v 2 ) θ(u 1, v 2 ) θ(u 2, v 1 ) + θ(u 1, v 1 ) = α 3 (π α 4 ) (π α 2 ) + α 3 = < 2π. 4 α i 2π We now shift gears from local considerations to global ones and define a map x : 2 S, which we will show to be a parameterization of S, as follows. Fix a point O S. Since S has negative curvature, there are precisely two asymptotic curves in S passing through O. We choose arbitrary orientations for these two curves and denote them a 1 and a 2. Now, given (s, t) 2, traverse curve a 1 from O for a distance s to arrive at a new point, p. There are again exactly two asymptotic curves passing through p, one of which is of course a 1. We denote the other by ã 2. We give ã 2 the orientation obtained by continuous extension along a 1 of the orientation of a 2. Then, traveling along ã 2 for a distance t, we arrive at our final destination, which we define to be x(s, t). We must show that x(s, t) is indeed defined for all (s, t) 2. First suppose that x(s, 0) is not defined for all s. Then there is a first s 1 such that x(s 1, 0) is not defined, i.e., an s 1 for which x(s 1, 0) is not defined but x(s, 0) is defined for s < s 1. Since x(s, 0) is parameterized by arc length, the sequence (a 1 (s)) is Cauchy as s s 1 and therefore converges to some q S by completeness of S. By Lemma 2, then, a 1 (s 1 ) is defined, contradicting our choice of s 1. The proof that x(s, t) is defined for all s and t is analogous. i=1

3 HILBET S THEOEM ON THE HYPEBOLIC PLANE 3 Lemma 4. For fixed t, the curve s x(s, t) is an asymptotic curve with s as arc length. Similarly, for fixed s, the curve t x(s, t) is asymptotic curve with t as arc length. Proof. We certainly have that s x(s, 0) is an asymptotic curve by construction. From the proof of Lemma 2, we have for sufficiently small δ that x(s, δ) can arrived at by first traversing the asymptotic curve α 2 from p (in the notation used in defining x) for a distance δ to a point q and then traversing the other asymptotic curve ã 1 through q for a distance s, i.e., by reversing the order of traversal from the definition of s. We still arrive at x(s, δ) because the asymptotic curves a 1, ã 1, a 2, and ã 2 locally form a Tchebyshef net. To get the result for all t, we cover the compact segment x(s, t ), 0 t t, with a finite number of Tchebyshef rectangles and appeal to our small-t result. The case of curves t x(s, t) for fixed s is completely analogous. Lemma 5. x is a local diffeomorphism. Proof. From Lemma 4, we have that x(s 0, t) and x(s, t 0 ) are asymptotic curves parameterized by arc length. By Lemma 2, we can locally parameterize S in such a way that the coordinate curves are the asymptotic curves of S and form a Tchebyshef net, i.e., have E = 1 = G. Then x must agree with this local parameterization. Since a parameterization is a diffeomorphism by definition, x is a local diffeomorphism. Lemma 6. x is surjective. Proof. Let Q = x( 2 ). Then Q is open in S, as x is a local diffeomorphism. Further, if p Q, then the two asymptotic curves passing through p are entirely contained in Q. We now assume for a contradiction that Q S. Since S is connected and is the disjoint union S = Int(Q) Bd(Q) Ext(Q) with Int(Q) and Ext(Q) open, we must have Bd(Q). Let p Bd(Q). Then p Q. Let be a neighborhood of p as described in Lemma 2 and let q Q. (Such a q exists by definition of the boundary of Q.) Then one of the asymptotic curves through q intersects one of the asymptotic curves through p, allowing us to extend the definition of x outside of Q, a contradiction. Lemma 7. There are two differentiable, linearly independent vector fields on S that are tangent to the asymptotic curves of S. Proof. Intuitively, at any point p S, there are four asymptotic directions: ±v 1 and ±v 2. We must make a consistent choice of sign in order to define two differentiable, linearly independent vector fields V 1 and V 2 on all of S. Basically, given a point q S, we take any curve α connecting p and q and let V i (q) be the continuous extension of v i along α. We then show that this definition is independent of the choice of curve α. We leave the technical, analytical details to do Carmo, pausing to note only one: the proof makes use of a path homotopy between α and any other choice of pq-curve α. That is, the proof relies on the simpleconnectedness of S, which actually is not terribly surprising, given the well-known topological obstructions to the existence of global vector fields on certain spaces. Lemma 8. x is injective. Proof. Lemma 7 yields linearly independent vector fields V 1 and V 2 that are everywhere tangent to the asymptotic directions. Then, from Lemma 4, x t (t, s) = V 1(x(t, s)), x s (t, s) = V 2(x(t, s)), i.e., the curves t x(t, s) for fixed s and s x(t, s) for fixed t are flow lines of the vector fields V 1 and V 2, respectively. (We choose, without loss of generality, the fixed s curves to correspond to V 1 and the fixed t curves to correspond to V 2 and not vice versa.) We refer to these collections of curves as Family 1 and Family 2, respectively. It follows from the usual uniqueness theorem of ODEs that curves within the same family cannot cross. Further, curves from different families can cross at most once; indeed, suppose C 1 is a curve of Family 1 and C 2 is a curve of Family 2 such that C 1 and C 2 cross at least twice. Then at least one of the curves, say C 1 must turn around : there is a point q S at which C 1 is tangent to another Family 2 curve, C2. But this contradicts that there are two linearly independent asymptotic directions at q. The restrictions on curve crossings within and among families then gives the injectivity of x.

4 4 MATTHEW D. BOWN The reader following along in do Carmo may notice that do Carmo gives a different (and longer) proof of Lemma 8 that makes a fair amount of use that S is topologically planar and not cylindrical or toroidal. One may wonder, then, how we managed to dispense with this. In fact, we did not: to prove the existence of the globally-defined vector fields V 1 and V 2 of Lemma 7, we had to utilize the fact that S is simply-connected. We can now deliver the coup de grâce: Proof of Main Theorem. We, as before, assume for a contradiction that there is an isometric immersion ψ : S 3 of the complete surface S with K 1 into 3. We choose an arbitrary p S and let S be the tangent plane T p S with the metric induced by the exponential map exp p : T p S S, which is a local diffeomorphism. Then ϕ exp p : S 3 is an isometric immersion. By Lemmas 4, 6, and 8, there is a global parameterization (diffeomorphism) x : 2 S whose coordinate curves are asymptotic curves of S. Therefore, we can cover S by coordinate quadrilaterals Q n := x ([ n, n] [ n, n]) with Q n Q n+1 for n N. This implies that an obvious contradiction. = Area(S ) (by Lemma 1) = lim n Area(Q n) 2π (by Lemma 3), 2. Appendix: Some Differential Geometry of Curves and Surfaces In this appendix, we give a brief account of the tools from differential geometry that we need to understand and prove Hilbert s Theorem. We assume that the reader (like the author) is more comfortable with the intrinsic properties of abstract manifolds than the extrinsic ones of curves and surfaces in 3, so we place emphasis on the latter Surfaces. A map ϕ : S 3 is called an immersion if its differential dϕ p : T p (S) T ϕ(p) ( 3 ) is injective for all p S. We call ϕ an isometric immersion if it further satisfies the condition (1) dϕ p (v), dϕ p (w) ϕ(p) = v, w p, p S, v, w T p (S). Observe that condition (1) actually implies that ϕ is an immersion (since dϕ p (v) = 0 implies that 0 = dϕ p (v) ϕ(p) = v p ), so that one might object that the term isometric immersion is redundant and could be replaced with, say, isometry. However, to keep with the conventions of many authors, we will reserve the term isomtery to refer to a diffeomorphism ϕ satisfying (1). A regular surface is a two-dimensional immersed submanifold of 3. Explicitly, a subset S 3 is a regular surface if, for each p S, there exists a neighborhodd V 3 and a map x : U V S of an open set U 2 onto V S 3 such that (1) x is smooth, i.e., if we write x(u, v) = (x(u, v), y(u, v), z(u, v)) for (u, v) U, then the functions x, y, z have continuous partial derivatives of all orders on all of U. (2) x is a homeomorphism. (3) x is an immersion. The map x above is referred to as a local parameterization or a local coordinate system. Note that property (3) implies that x u (q) = dx q(e 1 ) and and x v (q) = dx q(e 2 ) form a basis of T p S for every p V S, where e 1 and e 2 are the standard basis vectors of 2 and q = x 1 (p). The map u x(u, v 0 ) for (u, v 0 ) U is called the coordinate curve v = v 0. An abstract surface is a two-dimensional smooth manifold without boundary. A geometric surface is a two-dimensional iemannian manifold without boundary The First Fundamental Form. Given a regular surface S, the inner product of 3 induces an inner product, p on each tangent plane T p S for p S. The associated quadratic form I p given by I p (w) = w, w for w T p S is called the first fundamental form of S at p S. Given a local parameterization x : U V S of S, we define smooth functions E, F, G on U by E = x u, x u, F = x u, x v, G = x v, x v.

5 HILBET S THEOEM ON THE HYPEBOLIC PLANE 5 Since x u (q) and x v (q) form a basis for T p S for every p S V (where again q = x 1 (p), E, F, and G completely specify the first fundamental form locally. For concreteness, given p S V, let v T p S) and q = x 1 (p). If t (u(t), v(t)) is a curve in U such that (u(0), v(0)) = q and d dt t=0 x(u(t), v(t)) = v, then I p (v) = x u (q)u (0) + x v (q)v (0), x u (q)u (0) + x v (q)v (0) p = E(q) (u (0)) 2 + 2F (q)u (0)v (0) + G(q) (v (0)) 2 (Note that, given v T p S, such a curve t (u(t), v(t)) always exists.) This gives rise to the iemannianreminiscent line-element formula ds 2 = E du 2 + 2F du dv + G dv 2. Indeed, the first fundamental form is just the quadratic form associated to the metric induced on S by the inclusion map into 3. It can be shown that the area of a coordinate region of a surface S is given by Area() = EG F 2 du dv The Second Fundamental Form and Curvature. Let S be a regular surface. An orientation of S is a differentiable map N : S 3 such that N(p) is a unit vector orthogonal to p for every p S. S is said to be orientable if it admits an orientation. We can view N as a map S S 2, the unit sphere in 3, in which case we call N the Gauss map of S. Since T p S and T N(p) S 2 are parallel planes in 3, we can view the differential dn p for p S to be an endomorphism of T p S. It is not too hard to show that the differential dn p : T p S T p S is a self-adjoint linear map (with respect to the standard inner product on 3 T p S). To the self-adjoint linear map dn p we associate a quadratic form II p, called the second fundamental form of S at p, defined by II p (v) = dn p (v), v. Given a regular curve C in S parameterized by α(s), where s is the arc length of C, one can show that II p (α (0)) = k n (p) where α(0) = p and k n (p) = α (s) is the normal curvature of C at p. One consequence of this result is that the normal curvature of a curve in S at a point p S depends only on p and the tangent vector of the curve at p. Thus, we may speak of the normal curvature along a given direction at p. We may consider the maximum and minimum normal curvatures k 1 and k 2 at p and define the Gaussian curvature K to be K = k 1 k 2. We may write the second fundamental form in local coordinates as where II p (α ) = e (u ) 2 + 2f u v + g (v ) 2, e = N u, x u = N, x uu, f = N v, x u = N, x uv = N, x vu = N u, x v, g = N v, x v = N, x vv. It can be shown that, in this notation, the Gaussian curvature is given by K = eg f 2 EG F Tchebyshef Nets. The coordinate curves of a local parameterization x(u, v) of a surface S form a Tchebyshef net if the lengths of the opposite sides of any quadrilateral formed by them are equal. One can show that this is equivalent to E v = 0 = G u (where subscripts denote partial derivatives) and, if this is the case, it is possible to reparameterize the coordinate neighborhood in so that the coefficients of the first fundamental form are given by E = 1 = G and G = cos θ, where θ is the angle formed by coordinate curves. Further, it can be shown that, in such a parameterization, we have (2) K = θ uv sin θ.

6 6 MATTHEW D. BOWN 2.5. Asymptotic Curves. An asymptotic direction of S at p to be a direction in T p S along which the normal curvature is zero. There are exactly two asymptotic directions (up to a choice of sign) at any point of S that has strictly negative Gaussian curvature. (This essentially follows from the negative definiteness of the second fundamental form.) An asymptotic curve is a regular, connected curve in S whose tangent vector is an asymptotic direction at every point. That is, it is a curve whose normal curvature is everywhere zero. It can be shown that a necessary and sufficient condition for a curve on a surface of strictly negative Gaussian curvature to be an asymptotic curve is that e = 0 = g Miscellanious. We require the following two theorems for our proof of Hilbert s Theorem. For the first, see [5], p For the second, see [3], p Theorem (Hadamard). If M and M are connected iemannian manifolds with M complete and π : M M is a local isometry, then M is complete and π is a covering map. Theorem (Minding). Any two regular surfaces with the same constant Gaussian curvature are locally isometric. This is actually a consequence of the more general result that any two iemannian manifolds with the same constant sectional curvature are locally isometric (see [5], p. 181), as the sectional curvature in dimension two is just the Gaussian curvature. eferences [1] D. Hilbert. Grundlagen der Geometrie. Teubner, [2] D. Hilbert. Über Flächen von konstanter Gausscher Krümung. Trans. Amer. Math. Soc. 2 (1901), [3] M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, [4] A. Treibergs. The Hyperbolic Plane and its Immersions into 3. Unpublished notes, [5] J. M. Lee. Introduction to iemannian Manifolds. Springer, 1997.